ia-13inverse trigonometric functions (60-64)
TRANSCRIPT
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13. IN VERSE TRIGONOMETRIC FUNCTIONS
Synopsis :
1. The function f : [/2, /2] [1, 1] defined by f(x) = sinx is a bijection. The inverse of f from[1, 1] into [/2, /2] is also a bijection. This function is called inverse sine function or arc sine
function. It is denoted by Sin1 or Arc sin.
Now Sin1x = x = sin, x [1, 1]
2. The domains and ranges of the inverse trigonometric functions are as follows.
S.No Function Domain Range
1. Sin1x [1, 1] [/2, /2]
2. Cos1x [1, 1] [0, ]
3. Tan1x R (/2, /2)
4. Cot1x R (0, )
5. Sec1x (, 1] U [1, ) =
R(1, 1)
[0, /2) U (/2, )
6. Cosec1
x (, 1] U [1, ) =
RR(1, 1)
[/2, 0) U (0, /2)
3. i) Sin1(x) = Sin1x, for x [1, 1]
ii) Cos1(x) = Cos1x, for x [1, 1]
iii) Tan1(x) = Tan1x, for x R
4. i) Cot1(x) = Cot1x, for x R
ii) Sec1(x) = Sec1x, for (, 1] U [1, )
iii) Cosec1
(x) = Cosec1x, for (, 1] U [1, )
5. i) Sin1(x) = Cosec1(1/x), for x [1, 0) U (0, 1]
ii) Cos1
(x) = Sec1
(1/x), for x [1, 0) U (0, 1]
iii) Tan1(x) = Cot1(1/x), for x (0, ) and
Tan1x = + Cot1(1/x), for x (, 0)
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Inverse Trigonometric Functions
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6. If x [1, 1] then Cos1x + Sin1x = /2.
7. If x R, then Tan1x + Cot1x = /2.
8. If x
(
,
1] U [1,
), thenSec
1x + Cosec
1x = /2
9. If 0x1, 0y0, then
i) Sin1x + Sin1y = )x1yy1x(Sin 221 + , for x2 + y21
ii) Sin1
x + Sin1
y = )x1yy1x(Sin 221 + , for x2 + y2>1
10. If 0x1, 0y1, then Sin1xSin1y = )x1yy1x(Sin 221
11. If 0x1, 0y1, then
i) Cos1x + Cos1y = )y1x1xy(Cos 221 , for x2 + y21
ii) Cos1
x + Cos1
y = )xyy1x1(Cos 221 , for x2 + y2>1
12. If 0x1, 0y1, then Cos1xCos1y = )y1x1xy(Cos 221 +
13. Tan1x+Tan1y = Tan1
+
xy1
yx, for x>0, y>0, xy0, y > 0, xy > 1.
14. Tan1x+Tan1y = Tan1
+
xy1
yx, for x0, then Tan1xTan1y = Tan1
+
xy1
yx.
16. If x
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18. i) If Cos1x + Cos1y + Cos1z = , then
x2 + y2 + z2 + 2xyz = 1.
ii) If Sin1x + Sin1y + Sin1z = /2, then
x2
+ y2
+ z2
+ 2xyz = 1.
iii) If Sin1x + Sin1y + Sin1z = , then 222 z1zy1yx1x ++ = 2xyz.
iv)If Tan1x + Tan1y + Tan1z = /2, then
xy + yz + zx = 1.
v)If Tan1x + Tan1y + Tan1z = , then
x + y + z = xyz.
19. i) If Sin1x + Sin1y = , then Cos1x + Cos1y =
ii) If Cos1x+ Cos1y = , then Sin1x + Sin1y =
20. i) If Tan1x + Tan1y = /2, then xy = 1.
ii) If Cot1x + Cot
1y = /2, then xy = 1.
21. i) If 2211 baxthen,2x
bSin
x
aSin +=
=+
ii) If 2211 baxthen,2x
bCos
x
aCos +=
=+
iii) If baxthen,2x
bTan
x
aTan 11 =
=+
22. If then,b
yCos
a
xCos 11 =+ =+ 2
2
2
2
2
sinb
ycos
ab
xy2
a
x
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